×

zbMATH — the first resource for mathematics

Convexifactors, generalized convexity, and optimality conditions. (English) Zbl 1172.90500
The authors make use of the concept of convexificators which was introduced by V. Jeyakumar and D. T. Luc [J. Optimization Theory Appl. 101, No. 3, 599–621 (1999; Zbl 0956.90033)] to establish some characterizations of quasiconvex and pseudoconvex functions. They prove also a rule for calculating a convexificator of a max function, and derive optimality conditions for inequality constrained programming problems.

MSC:
90C46 Optimality conditions and duality in mathematical programming
90C26 Nonconvex programming, global optimization
68Q25 Analysis of algorithms and problem complexity
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] JEYAKUMAR, V., and LUC, D. T., Nonsmooth Calculus, Minimality, and Monotonicity of Convexifactors, Journal of Optimization Theory and Applications, Vol. 101, pp. 599-621, 1999. · Zbl 0956.90033 · doi:10.1023/A:1021790120780
[2] WANG, X., and JEYAKUMAR, V., A Sharp Lagrangian Multiplier Rule for Nonsmooth Mathematical Programming Problems Involving Equality Constraints, SIAM Journal on Optimization, Vol. 10, pp. 1136-1148, 2000. · Zbl 1047.90075 · doi:10.1137/S1052623499354540
[3] CLARKE, F. H., Optimization and Nonsmooth Analysis, Wiley Interscience, New York, NY, 1983. · Zbl 0582.49001
[4] MICHEL, P., and PENOT, J. P., A Generalized Derivative for Calm and Stable Functions, Differential and Integral Equations, Vol. 5, pp. 433-454, 1992. · Zbl 0787.49007
[5] CRAVEN, B. D., RALPH, D., and GLOVER, B. M., Small Convex-Valued Subdifferentials in Mathematical Programming, Optimization, Vol. 32, pp. 1-21, 1995. · Zbl 0816.49007 · doi:10.1080/02331939508844032
[6] MARTINEZ-LEGAZ, J. E., and SACH, P. H., A New Subdifferential in Quasiconvex Analysis, Journal of Convex Analysis, Vol. 6, pp. 1-11, 1999. · Zbl 0942.49020
[7] LUC, D. T., On Generalized Convex Nonsmooth Functions, Bulletin of the Australian Mathematical Society, Vol. 49, pp. 139-149, 1994. · Zbl 0811.90096 · doi:10.1017/S000497270001618X
[8] ORTEGA, J. M., and RHEINBOLT, W. C., Iterative Solutions of Nonlinear Equations, Academic Press, New York, NY, 1970.
[9] PENOT, J. P., and SACH, P. H., Generalized Monotonicity of Subdifferentials and Generalized Convexity, Journal of Optimization Theory and Applications, Vol. 94, pp. 251-262, 1997. · Zbl 0899.49007 · doi:10.1023/A:1022628223741
[10] LUC, D. T., The Characterizations of Quasiconvex Functions, Bulletin of the Australian Mathematical Society, Vol. 48, pp. 393-406, 1993. · Zbl 0790.49015 · doi:10.1017/S0004972700015859
[11] AUSSEL, D., Subdifferential Properties of Quasiconvex and Pseudoconvex Functions: Unified Approach, Journal of Optimization Theory and Applications, Vol. 97, pp. 29-45, 1998. · Zbl 0911.90295 · doi:10.1023/A:1022618915698
[12] LUC, D. T., Generalized Monotone Set-Valued Maps and Bifunctions, Acta Mathematica Vietnamica, Vol. 21, pp. 212-252, 1996. · Zbl 0893.47034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.